Bruce: In BioCalc, particularly, I've noticed the following pattern emerge. Typically a female student, after several weeks in the program, will walk up to me and express concern about the fact that she and her boyfriend--who's in engineering and is in the standard calculus course--were working on their homework together and it seems that he's learning and doing things that she's not learning. Therefore, she thinks she's not getting what she needs from the course. I look at her and I say, "Why do you automatically assume that you're not getting what you need? Why aren't you questioning whether he's getting what he needs?" I don't do anything but say, "Look at it from the other standpoint. You're learning all kinds of things." Sometimes I'll talk about the things that she's learning that she may not even realize are part of the learning process: learning to think about a problem, learning to set up a problem--all these things that are not answer oriented but are process oriented. Then, typically after a month or so, that student never fails to come back to me and say, "You know, my boyfriend was working on his homework, and I was able to help him because I understood what the point of the problem was. I understood the process that needed to go on, but he was too mired in the details. He was too focused on pushing symbols around." And at that point, it's like a fire gets lit under them.

The activities and tools chosen by the bricoleurs (described in The Learning Environment) help set in motion the processes that light a fire under their students. Their students are not just learning math, Jerry told us; they're learning how to learn.

Of note, the BioCalc learning environment is in strong accord with the seven principles for good practice in undergraduate education that Gamson and Chickering synthesized from their research on undergraduate education (1991). Below, we present each principle, followed by examples and testimony of how that principle is enacted in the BioCalc learning environment.

Good practice encourages contact between students and faculty.
"Frequent student-faculty contact in and out of class is a most important factor in student motivation and involvement. Faculty concern helps students get through rough times and keep on working. Knowing a few faculty members well enhances students' intellectual commitment and encourages them to think about their own values and plans."

The UIUC faculty place a high emphasis on contact with students. The result is an environment that is less formal and much more communicative than a typical, lecture-based course.

Brad: One of the things that's really wonderful about the course, in my opinion, is that it is not in the formal voice of traditional textbooks, and that encourages the students to respond. I've had students go so far as to turn in homework assignments that had their names at the top and then my name as the instructor appended with the words "Rock Star" because they thought something about the way I was acting needed to be lampooned in this manner. You do not see that on your homework in traditional class. Here, the students interact with you in their homework in a way that is much more like the way their own mind works.

One of the factors in establishing a more communicative environment is class size. BioCalc and C&M courses have typical enrollments of 15-30 students. As one student noted, "I like the smaller atmosphere, because I can get help whenever I need it. We can always get our questions answered, and no one has to wait." Being able to approach the instructor in class, this student also commented, is easier and less intimidating "than in the traditional course, where you'd have to make an appointment or find out the instructor's office hours." In BioCalc, another student told us, "you just raise your hand and [the instructor] comes over." Communication is further facilitated by the lab environment, where students and instructors spend eighty percent of their class time.

Luca* (graduate teaching assistant): You're talking to them personally most of the time. They're sitting in groups of two or maybe three in front of the computer. They ask a few questions, you come to them; you talk to them a lot during lab. In the traditional, lecture-based section, you might not talk to your students at all during the whole semester. You don't have even a chance to meet your students. In BioCalc, that's impossible. Even if a student does not ask a question, you can ask them if everything's okay, you can ask how it's going, you can ask some questions to make sure they understand what they're doing.

Moreover, student-faculty contact is made easier and more effective by the availability of a course website ("Course Space"). Here, students can post messages to their instructors, email them, or even chat with them online.

Good practice develops reciprocity and cooperation among students.
"Learning is enhanced when it is more like a team effort than a solo race. Good learning, like good work, is collaborative and social, not competitive and isolated. Working with others often increases involvement in learning. Sharing one's ideas and responding to others' improves thinking and deepens understanding."

Collaborative learning is another key feature of BioCalc and C&M courses. Students are strongly encouraged to work together, both in class and out, because the bricoleurs believe students learn better in a social and collaborative environment. And, Jerry noted, "most of the time you do see students working together--to get an idea of what a derivative is, for example."

Many instructors allow students to choose their own partners, but Bruce prefers a "mix-and-match" approach, in order to amplify results.

Bruce: There are two reasons why I mix and match groups. One reason is so people don't get stuck [with an unsuitable partner]. The other is that, for the first lesson they're partnered with someone and then the next lesson they're partnered with someone else, and then the next lesson they're partnered with someone else... Then this interesting multiple group thing starts to happen where a group is working on a problem and they get stuck. One of them says, "Oh, I'll go ask so-and-so, who I worked with last time." So the groups start grouping themselves together, and I can tell it's clicking when somebody jumps up from their computer and runs over to another group. That's a clicking indicator: when they actually start talking--not just among their group--but with other groups.

The students, too, are quick to pick up on the benefits of a more cooperative environment. Ann and Michelle, two students whom we interviewed, explained:

Ann: Michelle explains it to me when I don't understand.

Michelle: I think it's good because if there's something that I kind of get, by explaining it back to her and making sure I understand it--because she questions me on anything--if I didn't understand it entirely myself, between the two of us, it makes us go back and have to check and make sure that we understand everything that's going on. We can't just look at the computer and be, like, "Oh, okay, I got that," and go on. We have to explain it to each other.

They also feel free to consult others in the class.

Michelle: If we don't understand something, we'll ask around and find somebody else who's either on the same problem or has just worked it.

Creating a less formal environment encourages not only more faculty-student interaction, but more student-student interaction, as well.

Dan* (graduate teaching assistant): Bio-calc meets five days a week, so the students see each other every day, and they know each other very well. I actually have a student who brings a box of drinks every week, every Thursday. And when one of the students had a birthday, we actually celebrated in class. We benefit from stuff like that, actually, because the students won't hesitate when they are trying to help the other. They'll just go ahead and say, "Oh, I know that," and just start talking to each other instead of being shy.

Being in a calculus class with other life science students is an appealing feature of BioCalc, another student told us.

Steve: Well, I knew Mathematica was a new program that they were trying out here. I heard it was a math class geared more towards life science majors. So I thought maybe that would better suit me, rather than the normal calculus class. Plus, there was the advantage of meeting other people that were going into the same major as I was. It was neat because I actually got to be good friends with some of the people in that class and they were later in my lab classes. So, it was kind of a nice starting block.

BioCalc was formulated in part to offer a more supportive environment to students. As Susan Fahrbach, director the Howard Hughes Program in Life Sciences, noted, the power of learning in a social context is often overlooked.

Susan: I think this is a key point--that those students who like BioCalc enjoy being in a class with other life science majors. They're meeting people that they're going to be in classes with for the rest of their time at Illinois. I think we tend to underestimate that social aspect.

But students don't just talk and work together in class. They also use the Course Space website to email, chat and post messages to one another.

Good practice uses active learning techniques.
"Learning is not a spectator sport. Students do not learn much just sitting in classes listening to teachers, memorizing prepackaged assignments, and spitting out answers. They must talk about what they are learning, write reflectively about it, relate it to past experiences, and apply it to their daily lives. They must make what they learn part of themselves."

Active learning is the backbone of BioCalc and other C&M courses. As Jerry put it, "Math is not a spectator sport. You don't learn math by watching someone else do it."

We've already presented the bricoleurs ideas on how the graphics-based, interactive C&M notebooks more actively engage students in their own learning (The Learning Process). But the students, too, recognize how the BioCalc learning environment induces them to learn in a more active way.

In the notebooks, for examples, students are able to rework and restyle problems to suit their own needs, as many times as they need. This allows them to stay engaged with a problem for as long as they want and then move on, rather than having to work a preset number of problems.

Michelle: I prefer working with the Mathematica. I found the other way was just regurgitation-style. Here, you have to actually figure things out. The computer gives you the answer [in the Basics and Tutorial sections], but you have to figure out what it did--on your own. I find this a lot more interesting than doing the same type of problems twenty or thirty times just to get the steps down.

Another feature of the notebooks that the bricoleurs believe promotes active learning is the word processing: asking students to give written explanations of their answers gives them the opportunity to organize their thoughts and reinforces learning.

Bruce: The problems in the standard [lecture-based calculus] are algorithmic and fact-oriented. In this class, we're more process-oriented, very communication-oriented. We want to develop conceptual understanding. The text feature of Mathematica makes it easier to do that because the questions aren't: "Find the answer to this," or "Do this algorithm to get this answer." It's: "Why does that work? Look at this picture and explain in your own words what this picture tells you."

Moreover, the students themselves told us that the interaction required by the C&M courseware makes them feel more productive.

Elizabeth: This is more interactive. I was more passive in the last class--I just sat there and didn't really participate that much, because there wasn't really much participation to do obviously in the lectures. We'd sit there for an hour. And then the discussion, you can ask questions if you want to, otherwise you sit there and take more notes. But here, I just feel like I'm doing more. I'm being more productive than I was last semester, and I feel like I'm learning more.

Good practice gives prompt feedback.
"Knowing what you know and don't know focuses your learning. In getting started, students need help in assessing their existing knowledge and competence. Then, in classes, students need frequent opportunities to perform and receive feedback on their performance. At various points during college, and at its end, students need chances to reflect on what they have learned, what they still need to know, and how they might assess themselves."

Learning is further facilitated by the immediate feedback students get while working through problems and examples. By changing input factors, students are able to manipulate and change output data, changes that they see immediately with the press of a button. As Dan, a graduate teaching assistant, explained, immediate feedback enables students to better understand ideas and gives them a sense of satisfaction, as well.

Dan: One reason [students learn better] is because of the way the Mathematica lessons are structured. They get a sense of satisfaction when they do the problems because they can see the results. For instance, they start with a real world problem, they work through it, and at the end they see the results: perhaps a graph or something. And from that, it clicks. They're getting fairly immediate feedback, and that's very important.

Students do not just get feedback from the notebooks, however. If a student is having problems with any part of the lessons, he or she has a number of opportunities to get immediate help. During class, the instructor, class assistant and other students are available. When not in class, students have available three C&M labs on campus, each of which is staffed during the day with lab assistants. Students can also get help when they log in to Course Space; here they can enter an online chat room and discuss problems with other students or get help, at specified times, from C&M staff.

Good practice emphasizes time on task.
"Time plus energy equals learning. Learning to use one's time well is critical for students and professionals alike. Allocating realistic amounts of time means effective learning for students and effective teaching for faculty."

Time on task is almost a built-in feature of BioCalc because it's what Jerry calls a "unified learning environment." Students are able to spend most of their time in lab, working with a single, unified medium: the electronic notebooks. This, Jerry stressed, allows them to avoid the distractions of switching mediums (from book to pencil to paper to calculator, for instance) and instead maintain focus on the task at hand.

Because students are able to spend most of their class time on assignments, they can more easily get help when needed and the instructor can more accurately gauge student progress and adjust assignments accordingly.

Good practice communicates high expectations.
"Expect more and you will get it. High expectations are important for everyone -- for the poorly prepared, for those unwilling to exert themselves, and for the bright and well motivated. Expecting students to perform well becomes a self-fulfilling prophecy."

The bricoleurs told us that one of their goals for student learning was to convince students that math is something they can do. They expect their students to learn and to succeed, and the students pick up on that. The impetus to interact and engage with the ideas presented come from the students themselves, who understand the implicit assumption that they are responsible for their own learning.

Jeremy* (student & class assistant): Homework is very important. You can sit and listen to however many lectures, but if you don't do it yourself, you're probably not going to learn it. And when you get to Mathematica, it's based primarily on you doing it yourself.

Good practice respects diverse talents and ways of learning.
"Many roads lead to learning. Different students bring different talents and styles to college. Brilliant students in a seminar might be all thumbs in a lab or studio; students rich in hands-on experience may not do so well with theory. Students need opportunities to show their talents and learn in ways that work for them. Then they can be pushed to learn in new ways that do not come so easily."

BioCalc and other C&M courses invoke different methods of learning by using the strong visualization capacities of Mathematica; by asking students to not just solve problems, but to set up, solve, and explain problems; and by providing students with a context in which concepts and applications appear connected and applicable to real world pursuits. Several students told us that they learn better in this type of learning environment.

Tim (student & lab tech): It's easier to see how [concepts] directly apply, rather than getting lost in the theories. That sometimes happened to me when I took book calc in high school. I'm a very visual learner, and there are so many graphics [with Mathematica]--it helps me a lot to just be able to see what's happening, to be able to throw out tons and tons of graphs without a whole lot of work, and I can look and see every step of the way what it's doing. I learn the basic concepts through seeing it in action. That's what the Calculus&Mathematica courses are all about.

Jeremy, a C&M student and classroom assistant, expressed similar thoughts:

One of the advantages of Mathematica is that it's a lot more visual. If I took [a course without Mathematica], I'd be looking at equations all the time. I mean, how are you going to graph a 3-D vector field in a handwritten course? In the book, you get a nice picture of one equation. But [using Mathematica] I can come up with a 3-D vector field, a picture of it, for every single equation that I have.

Moreover, BioCalc relies heavily on using life science examples and applications to show students what calculus "is good for."

Luca* (graduate teaching assistant): Some students, when the math is getting complicated--when it's difficult, when it's a lot of material, it's hard and it's getting boring--they think that it's not for them. They don't understand why they need to solve all these problems. Why do they need to remember all these rules if they are not going to apply them in their real life? So that's why in the Mathematica courses we're going from the other position. We are showing them problems that they may meet in real life, and we're showing them how to solve these problems--how to apply mathematics to solve these problems.

Such an approach allows students of all levels to more easily make the important connections that both the math and life science faculty at UIUC say are needed.